Boolean cuts on one side but adds to the other
This behavior is correct, this is how the Boolean modifier works.

The Boolean modifier works better with manifold meshes (which, looking at your pictures, your mesh is not). If the mesh is not manifold, it does some guesswork which is inconsistent sometimes. I would recommend making your mesh manifold first by completing it.

If you wish to create a cut and not an incision, duplicate the front faces of the object you used to cut the model, separate it from the cutting object to make a new object, select both the new object and the object to be cut into, go into Edit Mode, hit A  to select all faces of both objects, and go to Mesh > Knife Project.

You should now have a hole without the supporting geometry.